摘要
继续沿着与"认知"、"数学哲学"、"无穷"相关的研究思路,结合新发现的集合论基础中所存在的问题和具体案例,研究"集合""、集合中的元素"、"一一对应"这三个集合论中最原始、基本内容的"本体—形式"论性质.结果表明,由于现有经典数学哲学、认知理论体系和无穷理论体系基础理论中的缺陷,导致集合论自创立以来,其基础理论中非常重要的一半内容一直处于完全空白的状态,使人们一直无法从"本体—形式"的角度认真对集合论的基础进行系统、科学的研究与认知[1-11],基础中的"缺失一半"问题是导致集合论中许多错误与悖论的真正根源,集合论基础中有一大片空白亟待人们去填补、开拓[12-16].在集合论基础的研究中,千百年来悬而未决的"无穷悖论综合症"揭示了人类科学现有理论体系中与"无穷"相关的基础理论深处肯定存在严重的缺陷,需要我们从很基础的地方去检查与研究,需要摆脱与"无穷"相关的现有理论体系中错误的"实无穷—潜无穷"思路的不良影响,解决这类特殊的疑难怪症需要开展"无污染基础研究"(这决定了该系列论文的文体和所引用参考文献的特殊性;应该根据所发现的集合类型,将各种各样的集合统一起来,构造出类似于"数谱"的"集合谱系",以"数谱"、"集合谱系"和"理论无穷—应用无穷"为基础[10],开拓全新的"集合分析"研究领域.
Going on the fundamental research thought relating to ＂cognition＂, ＂the philosophy of mathematics＂ and ＂infinitude＂, we naturally extent our work to the foundation of infinitude related set theory urged by the newly discovered fundamental problems and relating cases. The ＂ontology-form＂ natures of three primary and basic substances of ＂set＂, ＂the elements of set＂ and ＂one-to-one correspondence＂ are studied. A conclusion is drown：Because of the defects in the foundations of present classical cognitive theory, the philosophy of mathematics and infinite theory system, a half of the foundation of set theory has been totally absent thus unable people to study and understand systematically and scientifically the fundamental contents of set theory from the angle of ＂ontolo- gy-form＂ since set theory was founded [1-11]. The ＂half absence defect＂ in the foundation of present set theory is the real cause of mistakes and paradoxes in set theory. There is a huge blank space in the foundation of set theory waiting for making up and pioneering . Two new ideas are proposed： （ 1 ）The unsolvable ＂persistent syn- drome of infinitude related paradoxes＂ disclosed a fact that there must be serious defects deep in the foundation of present infinitude relating theory system and we should struggle off from the ill influences of the mistaken thoughts of ＂potential infinity and actual infinity＂ in the theory system, so checking and studying on the very foundation is a must and the work of ＂pollution-free fundamental research＂ is badly needed to solve such a special ＂paradox syn- drome＂ （which decides the specialties of the styles and the cited references of these serial papers）. （2）A ＂set spec- tnml＂, similar to the ＂number spectrum＂, should be constructed for all kinds of sets and a new field of ＂set analy- sis＂ based on ＂set spectrum＂, ＂number spectrum＂ and ＂theoretical infinitude-applied infinitude＂ should be opened up.
出处
《喀什师范学院学报》
2013年第3期29-34,共6页
Journal of Kashgar Teachers College
关键词
数学基础
数学哲学
无穷集合
集合中的元素
一一对应
“无污染”基础研究
集合谱系
集合分析
The foundation of mathematics
The philosophy of mathematics
Infinite set, The elements of set
One-to-one correspondence
＂Pollution-free＂ fundamental research
Set spectrum
Set analysis
